OPEN SUBJECT AREAS: MAGNETIC PROPERTIES AND MATERIALS ELECTRONIC DEVICES
Received 1 October 2013 Accepted 24 January 2014 Published 17 February 2014
Correspondence and requests for materials should be addressed to S.E.B. (barnes@ physics.miami.edu)
Rashba Spin-Orbit Anisotropy and the Electric Field Control of Magnetism Stewart E. Barnes1,2, Jun’ichi Ieda2,3 & Sadamichi Maekawa2,3 1
Physics Department, University of Miami, Coral Gables, FL 33124, USA, 2Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan, 3CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan.
The control of the magnetism of ultra-thin ferromagnetic layers using an electric field, rather than a current, has many potential technologically important applications. It is usually insisted that such control occurs via an electric field induced surface charge doping that modifies the magnetic anisotropy. However, it remains the case that a number of key experiments cannot be understood within such a scenario. Much studied is the spin-splitting of the conduction electrons of non-magnetic metals or semi-conductors due to the Rashba spin-orbit coupling. This reflects a large surface electric field. For a magnet, this same splitting is modified by the exchange field resulting in a large magnetic anisotropy energy via the Dzyaloshinskii-Moriya mechanism. This different, yet traditional, path to an electrically induced anisotropy energy can explain the electric field, thickness, and material dependence reported in many experiments.
T
he possibility of controlling the magnetic anisotropy of thin ferromagnetic films using a static electric field E is of great interest since it can potentially lead to magnetic random access memory (MRAM) devices which require less energy than spin-torque-transfer random access memory STT-MRAM1–7. Thin magnetic films with a perpendicular magnetic anisotropy (PMA) are important for applications8,9. That an interfacial internal electric field might be used to engineer such a PMA is also of great interest. Experiment10,11 has indeed shown that such a PMA might, in turn, be modified by an externally applied electric field, however the data is usually interpreted in terms of changes to the electronic contribution to magnetic anisotropy due to the surface doping induced by the applied electric field3,11,12. The theory of the field-induced changes of the magnetic anisotropy reflecting surface doping is invariably developed in terms of band theory13–17. The results for both the bulk and thin films can be adequately understood in terms of second order perturbation theory18 in which the relevant matrix elements of the spin-orbit interaction are between full and empty states. Large contributions come from regions where different d-bands (almost) cross. That such crossings should be close to the Fermi surface leads to a strong doping dependence in such theories. Nakamura et al.19 pointed out that the strong negative applied field dependence of the PMA for an isolated monolayer (ML) of Fe(001) arises directly from band splitting rather than from doping. In this case, an E perpendicular to the film breaks reflection symmetry causing a large spin-orbit splitting of d-levels near the Fermi surface. As will be explain below, despite these important theoretical developments, a clear explanation of a number of key experiments is still lacking. Here we develop a simple analytic theory for the existence and electrical control of the PMA based upon the Rashba spin-orbit interaction20–22 and the single band Stoner model of magnetism. We exhibit the somewhat delicate, but very interesting, competition between the Rashba spin-orbit fields and the exchange interaction, reflecting electron correlations. This theory can potentially lead to a very large magnetic anisotropy arising from the internal electric fields Eint which exist at, e.g., ferromagnetic/metal and ferromagnetic/oxide insulator interfaces but modified by the addition of an applied electric field Eext. There is a Rashba splitting of the band structure leading to a quadratic, (Eint 1 Eext)2, contribution to the magnetic anisotropy, contrasting with a linear in Eext doping effect.
Results Model. This comprises a band Stoner model with the Rashba interaction added23: H~
SCIENTIFIC REPORTS | 4 : 4105 | DOI: 10.1038/srep04105
p2 aR {J0 S:sz sx py {sy px : 2m h
ð1Þ
1
www.nature.com/scientificreports
(a)
BRysinθ
θ
y x BRx B ycosθ R
ˆ m
zˆ
R
F
ε
δ
reflects the electron image potential. For the Ag/Cu(111) system27 an E of similar magnitude occurs at the metal interface. The difference in the (111) chemical potentials28 of Cu(4.96 eV) and Ag(4.74 eV) is reflected by a potential increase from Cu to Ag. The chemical potential of Au(5.31 eV) implies a similar potential change and E between Cu and Au, but of the opposite sign. It is the s-electrons which penetrate the core and determine the spin-orbit parameter gso. The example Cu/Ag/Au therefore illustrates the higher/lower potential of these s-electrons in the second/third transition series relative to the similar electrons in the 3d elements. There are many experiments24,25,29–31 which put in evidence the Rashba splitting in two dimensional electron gases, surface states of noble metals, bulk layered systems, and e.g., of a surface state of ferromagnetic Yb.
BRy
EB
(c)
ky
(b) ε
z θ J0 S
kx
yˆ xˆ (d)
ky
ER
k0
ε
δ
ky
Magnetic case - origin of the magnetic anisotropy energy. In the methods it is shown the h dependent single particle energy, i.e., the equivalent of Eq. (2) for the magnetic case is:
kx
kx ˆ ˆ m||z
k0 -k0
2 h ðkx {sk0 sin hÞ2 zky 2 {ER sin2 h 2m 1=2 : {s ðJ0 SÞ2 za2R kx 2 cos2 hzky 2
ks ~
ˆ ˆ m||y
Figure 1 | (a) The electric field E~E^z is perpendicular to the ferromagnet ^ is defined by the angle h surface while the order parameter direction m, relative to ^z. Whatever the direction of k, the Rashba magnetic field BR of direction k 3 E lies in the ^ x{^y plane. (b) The Rashba split bands of a nonmagnetic metal. The two Fermi sheets emerge from a ‘‘Dirac point’’ near the bottom of the illustration. For the magnetic case the two Fermi sheets ^ the electron spins make a are disconnected. (c) For a perpendicular m constant angle d to the vertical such that the projection is as in (b). The ^ parallel to additional exchange splitting increases as E2. (d) Same but for m ^ along the y-direction the majority and minority Fermi the plane. With m seas shift along the x-axis in opposite directions. The tilt of the spin relative ^ is no longer a constant being zero along the x-axis and a maximum to m along the y-axis.
where p is the electron momentum operator, S the order parameter, s the Pauli matrices and aR 5 egsoE the Rashba parameter proportional to gso, which characterises the spin-orbit coupling. The electric field E~E^z is taken to be perpendicular to the plane of the system, ^ and m~S=S is perpendicular to ^ x and makes an angle h to the ^zdirection, as in Fig. 1(a). Illustrative non-magnetic example. Consider the Rashba effect in non-magnetic two dimensional electron gases or surface states on noble metals, e.g., a surface state of Au. As shown in the methods, the single particle energy 2 h ðk{sk0 Þ2 {ER , ks ~ 2m
ð2Þ
as the spin quantum number s 5 61. The momentum shift k0 ~maR h2 , and maR 2 1 egso 2 2 ~ mE , ð3Þ ER ~ 2 h 2 h2 identified in the inset of Fig. 1(b), reflects the single particle energy gain relative to zero electric field, i.e., E 5 0 and hence aR 5 0. For the h2 three dimensional problem ks ~ ðk{sk0 Þ2 zkz 2 {ER and 2m there is no equivalent momentum shift in kz. For the surface state of Au, ER < 3.5 meV24,25 exemplifying the energy scale. The study of such surface states, and differences in chemical potentials, also helps set the scale for E. Between a metal and the vacuum, or dielectric, the E , 10 V/nm near the Cu (100) surface26, SCIENTIFIC REPORTS | 4 : 4105 | DOI: 10.1038/srep04105
ð4Þ
The direction of the now h dependent momentum shift changes sign ^ ^ as the spin index s 5 61. These shifts also change sign with m?{ m ^ ^ is for a given s. This ‘‘magnetic Rashba splitting’’ with m?{ m observed for the surface state of Yb31. Also in the methods the Dzyaloshinskii-Moriya (DM) and pseudo-dipolar (PD) contributions to the magnetic anisotropy are highlighted by contrasting the perpendicular and parallel orienta^ to the plane. tions of order parameter m Assuming (J0S)2 . (aRkx)2 and retaining the h-dependent terms up to the order of E2 in (4), we obtained our principal result:
2T cos2 h, ð5Þ Ean ~ER 1{ J0 S for the magnetic anisotropy energy, with h2 2 2 T~ kx : { kx ; , 2m
ð6Þ
where Æ æ denotes an average over the Fermi sea (see methods). The Rashba spin-orbit interaction produces a uni-axial anisotropy energy which, as in the Dzyaloshinskii-Moriya theory32–35, comprises a direct second order in E easy plane pseudo-dipolar interaction and an indirect contribution proportional to E2/J0S reflecting the competition between the first order in E, Rashba-Dzyaloshinskii-Moriya, and exchange fields. Competition between the Dzyaloshinskii-Moriya and pseudodipolar contributions. Clearly Eq. (5) implies an E2 dependent PMA results if T . J0S/2, i.e., when the DM is larger than the PD term. Taken literally, the Stoner model Eq. (1), with its quadratic dispersion, predicts the ratio of the DM and PD contributions to the PMA. The result, (see methods), depends upon the spatial dimension. In two dimensions the PD and DM terms cancel although higher order terms (O(aR4)) lead to a PMA while in three dimensions the DM term is 2(4/5) ER cos2 h and an in-plane magnetisation is favoured. Lastly, a two dimensional system with a highly anisotropic conductivity might be modelled as a series of parallel one dimensional chains. For chains the DM contribution 2(4/3) ER cos2 h which dominates the PD energy ER cos2 h, ^ is in-plane and perpendicular to the chains. appropriate when m ^ in-plane but rather Corresponding to the hardest axis, when m parallel to the chains, there is neither a DM or PD contribution to the magnetic anisotropy energy. There is thereby a predicted electric field dependence of the in-plane anisotropy as seen in early 2
www.nature.com/scientificreports experiments3, given the large compressive strain that arises in these experiments. However, for the real problem of 3d magnets, a quadratic dispersion is not at all realistic and the crystal potential V(r) must be accounted for, see methods. For 3d elements the wavefunction y is well within the atomic sphere and the averages, e.g., localised h2 kx 2 , and hence T, are very much increased as compared to the above naı¨ve estimates. In reality, the DM contribution will invariably lead to a PMA.
electric field is increased at one surface and decreased at the other doubling the net effect. In contrast, for this same symmetric situation, the surface charges are opposite and doping effects must cancel. Experimentally applied fields of 1 V/nm are relatively easy to achieve implying a , 10% change in the surface anisotropy. Experiments40 with a 1.5 nm Fe80B20 sandwiched between two MgO layers are perhaps closest to this situation although the thickness 1.5 nm and 2.5 nm of these layers are not equal. Roughly consistent with our estimate [see Fig. 2(c), case(i)], there is40 an , 15% symmetric contribution to the magnetic anisotropy for an applied voltage of 2 V. Clearly the intrinsic Rashba field BR is modified when the materials adjacent to a 3d ferromagnet (F) are different. In a number of experiments an insulator I, often MgO, lies to one side and a normal metal (N), e.g., Au, Pt, Pd, Ta, or Ru, ‘‘cap’’ completes a tri-layer system. The potential, Figs. 2(d) and (e), will increase in passing from Fe to MgO but in passing from the F to N-layer the potential will either increase, Fig. 2(d), or decrease, Fig. 2(e). As discussed above, the relevant potential would be expected to increase, Fig. 2(d), in passing from a 3d element, e.g., Co, Fe, Ni, or Cu, to a 4d transition metal such as Ru, Pd, Ag but decrease, Fig. 2(e), for the 5d elements, e.g., Au, Pt, Ta. The latter case is particularly favourable since the intrinsic Rashba fields have the same sense and add. In addition, the 5d elements have a larger spin-orbit coupling, resulting in a larger aR and hence are more likely to produce a sizeable PMA. If the electric field decreases at the FI interface, the average Rashba field increases in the first case [Fig. 2(f)] when the effects of the surfaces tend to cancel and, as illustrated in Fig. 2(g), decreases in the second case when the inverse is true [see Fig. 2c, case (ii) and (iii)]. Experiment12 indeed shows an opposite field dependence for such systems with Pd(4d) and Pt(5d) N-layers. That the sign of the electric field contribution to the PMA reflects the N-layer whereas the field is applied to the opposite surface between F and I supports the current Rashba model. This is in stark contrast with the popular surface doping model3,12, for which the effects of surface doping are limited by the
Discussion The resulting anisotropy energy can be very large. The work already cited24–27 on conducting but non-magnetic materials helps set the scales. The value of the scaling prefactor ER in Eq. (5) for the surface state of Au is , 3.5 meV or about 35 T in magnetic field units and very much larger than the typical , 1 T demagnetising field. If a Au film is polarised by contact with an ultra-thin ferromagnet, the second factor, 2T/J0S, in Eq. (5) for the field inside a Au surface layer can be quite large , 5 leading to a PMA and indeed ultra-thin Fe on Au does have such a PMA36,37. Ultra-thin ferromagnetic films in contact with, e.g., Ru, Pd, Pt, and Ta, etc., also are found to have a PMA6,12,38,39. Schematically shown in Fig. 2(a) is the potential seen by electrons in a free standing ultrathin ferromagnetic film. At the surface, the potential reflects an electron’s image charge but reaches the vacuum level within a few atomic spacing. As already discussed, this results in a finite large electric field E , 10 V/nm at each surface but in opposite senses. Assuming an appreciable spin-orbit coupling in the interface region, this results, in turn, in a Rashba field BR which also changes sign between the two surfaces for a given momentum. Thus, for a perfectly symmetric film, the ferromagnetically polarised electrons see no average field BR. This symmetry can be broken by the application of an external electric field as shown in Fig. 2(b). The
/ va ins
-eV
BR
c
ro fer ins
c /v a
kx
V0 0 E
BR
(b)
+
c /v a ins
Vg
zˆ xˆ
yˆ
(d) /v ins
-eV
BR
-
E
ac ro fer ins/v
kx
kx
E BR
ins
BR E
E
ac
al ro fer met
ii)
Vg
c / va
kx
al ro fer met
+
(f) -eV
E +
(g)
-eV
ins
BR BR
E
kx
Vg
ins
BR
kx k x E BR
(e)
-eV BR
i) iii)
kx
E
-eV
Ean
(c)
(a)
E
-
c / va
a ro fer met
l
kx k x Vg
E BR
-
c /v a
al ro fer met
kx
BR E
kx
Figure 2 | (a) There is an electric field E in the surface region of a ferromagnet, however for a given wave vector k, the Rashba field BR, proportional to k 3 E, has an opposite sign at the two surfaces and the average field is zero. (b) With a finite external field this symmetry is broken and there is a net Rashba field acting upon the electrons. (c) The gate voltage dependence of the anisotropy energy. The internal electric field causes the shift of the parabola in the lateral axis as indicated by V0 for case i). For cases ii) and iii) the internal field shift is far beyond the external field range and nearly linear E-dependence arises. (d) The symmetry is also broken for a insulator-ferromagnet-metal sandwich. Also despite the electric field being smaller at the right surface, for a suitable metal, the spin-orbit coupling is larger and hence the metal interface can still dominate the net Rashba field. (e) Here the work function is larger for the metal than for the ferromagnet and the field for that surface is reversed. Now the Rashba fields at the two surfaces add. (f), (g) Applying positive gate voltages decreases the Rashba field at the insulating surface which, for this case, causes a net increase/decrease in the average Rashba field. SCIENTIFIC REPORTS | 4 : 4105 | DOI: 10.1038/srep04105
3
www.nature.com/scientificreports (possibly magnetically modified) Fermi-Thomas screening length. In reality12, the screening length is estimated to be much less than 1 nm, and much too short for there to be an appreciable doping effect of the Pt or Pd layers that are typically distant by a few nanometers. Simulating a large applied electric field E, the required asymmetry might be controlled in NFN tri-layers by varying in a systematic manner, at the mono-layer level, the thickness of one of the normal metal layers and by using metals with different spin-orbit couplings. In reality the effect of the substrate transmitted to, and through, the bottom normal metal will imply an asymmetry even for largish Nlayer thickness. Indeed the PMA surface term for Au/Fe(110)/ Au(111) structures does show an non-monotonic dependence on the top Au layer thickness36. Experiments37 for Fe layers on vicinal Ag(001) and Au(001) surfaces and which undergo a symmetry breaking (5 3 20) surface reconstruction manifest an in-plane surface term reflecting this broken symmetry and which is larger for Au, with its stronger spin-orbit coupling, than for Ag. It is predicted that the surface coercivity field Hc is proportional to (Eint 1 Eext)2 where Eint is the internal electric field corresponding to the zero-bias Rashba contribution to the anisotropy. Such a nonlinear field dependence is observed, e.g., for the in-plane contribution for a (Ge,Mn)As/ZrO2 surface3. In other experiments10 with CoFeB/ MgO/CoFeB structures there is qualitative difference between the E dependence of the anisotropy field Hc of the, ‘‘top’’ and ‘‘bottom’’, CoFeB layers of this three layer structure, even when they have similar thicknesses. The bottom layer has a larger Hc and is roughly linear while Hc becomes highly non-linear as Hc R 0 as would be expected as Eext R 2Eint. The most direct experimental test of the model is the observation of the band splittings for a model Rashba system with a variable contact with an itinerant ferromagnet. This can result in giant magnetic anisotropy (GMA) energies. For example an ER , 100 meV (or , 1000 T) is reported in angle-resolved photoemission spectroscopy (ARPES) measurements41 on bulk BiTeI. For a thin film of this, or similar material, in contact with an itinerant ferromagnet such as Fe, a suitable exchange splitting J0S, tuned to the order of ER, might be induced and a GMA will result. ARPES performed as a function of the direction of the magnetisation m might determine both ER and the momentum dependence of the exchange splitting leading to estimates of both the PM and DM contributions and which might be directly compared with magnetisation and magnetic resonance measurements. The electrical control of such a GMA has evident important application for non-volatile memory applications. There are clearly many more complicated embodiments of such a device. In conclusion, it is suggested that the Rashba magnetic field due to the internal electric field in the surface region of an ultra-thin ferromagnet can make an important contribution to the perpendicular magnetic anisotropy. Such surface fields might be modified by application of an applied electric field. Since the internal fields at two surfaces tend to cancel, an asymmetry between the surfaces is important. Such an asymmetry is caused by different metal and insulator caping layers. These ideas are consistent with a large number of experiments.
Methods The non-magnetic case. This corresponds to Eq. (1) with J0 5 0. It is solved by taking the axis of quantisation ^z|k tobeperpendicular to the in-plane k as in Fig. 1(b). The eigenstates are eik?rjsæ and H~ h2 2m k2 {gmB BR Ns=2, where the Rashba magnetic field in energy units is defined as gmB BR ~2aR {ky ^xzkx ^y , with mB the Bohr magneton and g the g-factor, leaving the spin state jsæ to be determined. There are two concentric The energy splitting 2aR k:Dðk=kF Þ, where D is the value Fermi surfaces. for kF : kF: zkF; 2, with kF",# the Fermi wave number for the spin up/down (s 5 61) band. For the surface state of Au, D < 110 meV while EF < 420 meV giving the ER < 3.5 meV cited in the text. The magnetic case. The full Eq. (1) is solved by ^ defining axes such that m:S=S lies in the y–z-plane and S~Sðcos h^zzsin h^yÞ. The ^ total field, which defines the axis of quantisation, gmB BT ~2½ðJ0 SzaR kx sin hÞm{ ^ xÞ. It is assumed that, for a 3d ferromagnet J0S , 0.5–1.0 eV xzaR kx cos hðm|^ aR k y ^ and gmBBR , J0S, i.e., the Rashba is smaller than the exchange splitting. To second
SCIENTIFIC REPORTS | 4 : 4105 | DOI: 10.1038/srep04105
^ where JS~ ðJ0 SÞ2 za2R kx 2 cos2 hz order in gmBBR, gmB BT
Received 1 October 2013 Accepted 24 January 2014 Published 17 February 2014
Correspondence and requests for materials should be addressed to S.E.B. (barnes@ physics.miami.edu)
Rashba Spin-Orbit Anisotropy and the Electric Field Control of Magnetism Stewart E. Barnes1,2, Jun’ichi Ieda2,3 & Sadamichi Maekawa2,3 1
Physics Department, University of Miami, Coral Gables, FL 33124, USA, 2Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan, 3CREST, Japan Science and Technology Agency, Sanbancho, Tokyo 102-0075, Japan.
The control of the magnetism of ultra-thin ferromagnetic layers using an electric field, rather than a current, has many potential technologically important applications. It is usually insisted that such control occurs via an electric field induced surface charge doping that modifies the magnetic anisotropy. However, it remains the case that a number of key experiments cannot be understood within such a scenario. Much studied is the spin-splitting of the conduction electrons of non-magnetic metals or semi-conductors due to the Rashba spin-orbit coupling. This reflects a large surface electric field. For a magnet, this same splitting is modified by the exchange field resulting in a large magnetic anisotropy energy via the Dzyaloshinskii-Moriya mechanism. This different, yet traditional, path to an electrically induced anisotropy energy can explain the electric field, thickness, and material dependence reported in many experiments.
T
he possibility of controlling the magnetic anisotropy of thin ferromagnetic films using a static electric field E is of great interest since it can potentially lead to magnetic random access memory (MRAM) devices which require less energy than spin-torque-transfer random access memory STT-MRAM1–7. Thin magnetic films with a perpendicular magnetic anisotropy (PMA) are important for applications8,9. That an interfacial internal electric field might be used to engineer such a PMA is also of great interest. Experiment10,11 has indeed shown that such a PMA might, in turn, be modified by an externally applied electric field, however the data is usually interpreted in terms of changes to the electronic contribution to magnetic anisotropy due to the surface doping induced by the applied electric field3,11,12. The theory of the field-induced changes of the magnetic anisotropy reflecting surface doping is invariably developed in terms of band theory13–17. The results for both the bulk and thin films can be adequately understood in terms of second order perturbation theory18 in which the relevant matrix elements of the spin-orbit interaction are between full and empty states. Large contributions come from regions where different d-bands (almost) cross. That such crossings should be close to the Fermi surface leads to a strong doping dependence in such theories. Nakamura et al.19 pointed out that the strong negative applied field dependence of the PMA for an isolated monolayer (ML) of Fe(001) arises directly from band splitting rather than from doping. In this case, an E perpendicular to the film breaks reflection symmetry causing a large spin-orbit splitting of d-levels near the Fermi surface. As will be explain below, despite these important theoretical developments, a clear explanation of a number of key experiments is still lacking. Here we develop a simple analytic theory for the existence and electrical control of the PMA based upon the Rashba spin-orbit interaction20–22 and the single band Stoner model of magnetism. We exhibit the somewhat delicate, but very interesting, competition between the Rashba spin-orbit fields and the exchange interaction, reflecting electron correlations. This theory can potentially lead to a very large magnetic anisotropy arising from the internal electric fields Eint which exist at, e.g., ferromagnetic/metal and ferromagnetic/oxide insulator interfaces but modified by the addition of an applied electric field Eext. There is a Rashba splitting of the band structure leading to a quadratic, (Eint 1 Eext)2, contribution to the magnetic anisotropy, contrasting with a linear in Eext doping effect.
Results Model. This comprises a band Stoner model with the Rashba interaction added23: H~
SCIENTIFIC REPORTS | 4 : 4105 | DOI: 10.1038/srep04105
p2 aR {J0 S:sz sx py {sy px : 2m h
ð1Þ
1
www.nature.com/scientificreports
(a)
BRysinθ
θ
y x BRx B ycosθ R
ˆ m
zˆ
R
F
ε
δ
reflects the electron image potential. For the Ag/Cu(111) system27 an E of similar magnitude occurs at the metal interface. The difference in the (111) chemical potentials28 of Cu(4.96 eV) and Ag(4.74 eV) is reflected by a potential increase from Cu to Ag. The chemical potential of Au(5.31 eV) implies a similar potential change and E between Cu and Au, but of the opposite sign. It is the s-electrons which penetrate the core and determine the spin-orbit parameter gso. The example Cu/Ag/Au therefore illustrates the higher/lower potential of these s-electrons in the second/third transition series relative to the similar electrons in the 3d elements. There are many experiments24,25,29–31 which put in evidence the Rashba splitting in two dimensional electron gases, surface states of noble metals, bulk layered systems, and e.g., of a surface state of ferromagnetic Yb.
BRy
EB
(c)
ky
(b) ε
z θ J0 S
kx
yˆ xˆ (d)
ky
ER
k0
ε
δ
ky
Magnetic case - origin of the magnetic anisotropy energy. In the methods it is shown the h dependent single particle energy, i.e., the equivalent of Eq. (2) for the magnetic case is:
kx
kx ˆ ˆ m||z
k0 -k0
2 h ðkx {sk0 sin hÞ2 zky 2 {ER sin2 h 2m 1=2 : {s ðJ0 SÞ2 za2R kx 2 cos2 hzky 2
ks ~
ˆ ˆ m||y
Figure 1 | (a) The electric field E~E^z is perpendicular to the ferromagnet ^ is defined by the angle h surface while the order parameter direction m, relative to ^z. Whatever the direction of k, the Rashba magnetic field BR of direction k 3 E lies in the ^ x{^y plane. (b) The Rashba split bands of a nonmagnetic metal. The two Fermi sheets emerge from a ‘‘Dirac point’’ near the bottom of the illustration. For the magnetic case the two Fermi sheets ^ the electron spins make a are disconnected. (c) For a perpendicular m constant angle d to the vertical such that the projection is as in (b). The ^ parallel to additional exchange splitting increases as E2. (d) Same but for m ^ along the y-direction the majority and minority Fermi the plane. With m seas shift along the x-axis in opposite directions. The tilt of the spin relative ^ is no longer a constant being zero along the x-axis and a maximum to m along the y-axis.
where p is the electron momentum operator, S the order parameter, s the Pauli matrices and aR 5 egsoE the Rashba parameter proportional to gso, which characterises the spin-orbit coupling. The electric field E~E^z is taken to be perpendicular to the plane of the system, ^ and m~S=S is perpendicular to ^ x and makes an angle h to the ^zdirection, as in Fig. 1(a). Illustrative non-magnetic example. Consider the Rashba effect in non-magnetic two dimensional electron gases or surface states on noble metals, e.g., a surface state of Au. As shown in the methods, the single particle energy 2 h ðk{sk0 Þ2 {ER , ks ~ 2m
ð2Þ
as the spin quantum number s 5 61. The momentum shift k0 ~maR h2 , and maR 2 1 egso 2 2 ~ mE , ð3Þ ER ~ 2 h 2 h2 identified in the inset of Fig. 1(b), reflects the single particle energy gain relative to zero electric field, i.e., E 5 0 and hence aR 5 0. For the h2 three dimensional problem ks ~ ðk{sk0 Þ2 zkz 2 {ER and 2m there is no equivalent momentum shift in kz. For the surface state of Au, ER < 3.5 meV24,25 exemplifying the energy scale. The study of such surface states, and differences in chemical potentials, also helps set the scale for E. Between a metal and the vacuum, or dielectric, the E , 10 V/nm near the Cu (100) surface26, SCIENTIFIC REPORTS | 4 : 4105 | DOI: 10.1038/srep04105
ð4Þ
The direction of the now h dependent momentum shift changes sign ^ ^ as the spin index s 5 61. These shifts also change sign with m?{ m ^ ^ is for a given s. This ‘‘magnetic Rashba splitting’’ with m?{ m observed for the surface state of Yb31. Also in the methods the Dzyaloshinskii-Moriya (DM) and pseudo-dipolar (PD) contributions to the magnetic anisotropy are highlighted by contrasting the perpendicular and parallel orienta^ to the plane. tions of order parameter m Assuming (J0S)2 . (aRkx)2 and retaining the h-dependent terms up to the order of E2 in (4), we obtained our principal result:
2T cos2 h, ð5Þ Ean ~ER 1{ J0 S for the magnetic anisotropy energy, with h2 2 2 T~ kx : { kx ; , 2m
ð6Þ
where Æ æ denotes an average over the Fermi sea (see methods). The Rashba spin-orbit interaction produces a uni-axial anisotropy energy which, as in the Dzyaloshinskii-Moriya theory32–35, comprises a direct second order in E easy plane pseudo-dipolar interaction and an indirect contribution proportional to E2/J0S reflecting the competition between the first order in E, Rashba-Dzyaloshinskii-Moriya, and exchange fields. Competition between the Dzyaloshinskii-Moriya and pseudodipolar contributions. Clearly Eq. (5) implies an E2 dependent PMA results if T . J0S/2, i.e., when the DM is larger than the PD term. Taken literally, the Stoner model Eq. (1), with its quadratic dispersion, predicts the ratio of the DM and PD contributions to the PMA. The result, (see methods), depends upon the spatial dimension. In two dimensions the PD and DM terms cancel although higher order terms (O(aR4)) lead to a PMA while in three dimensions the DM term is 2(4/5) ER cos2 h and an in-plane magnetisation is favoured. Lastly, a two dimensional system with a highly anisotropic conductivity might be modelled as a series of parallel one dimensional chains. For chains the DM contribution 2(4/3) ER cos2 h which dominates the PD energy ER cos2 h, ^ is in-plane and perpendicular to the chains. appropriate when m ^ in-plane but rather Corresponding to the hardest axis, when m parallel to the chains, there is neither a DM or PD contribution to the magnetic anisotropy energy. There is thereby a predicted electric field dependence of the in-plane anisotropy as seen in early 2
www.nature.com/scientificreports experiments3, given the large compressive strain that arises in these experiments. However, for the real problem of 3d magnets, a quadratic dispersion is not at all realistic and the crystal potential V(r) must be accounted for, see methods. For 3d elements the wavefunction y is well within the atomic sphere and the averages, e.g., localised h2 kx 2 , and hence T, are very much increased as compared to the above naı¨ve estimates. In reality, the DM contribution will invariably lead to a PMA.
electric field is increased at one surface and decreased at the other doubling the net effect. In contrast, for this same symmetric situation, the surface charges are opposite and doping effects must cancel. Experimentally applied fields of 1 V/nm are relatively easy to achieve implying a , 10% change in the surface anisotropy. Experiments40 with a 1.5 nm Fe80B20 sandwiched between two MgO layers are perhaps closest to this situation although the thickness 1.5 nm and 2.5 nm of these layers are not equal. Roughly consistent with our estimate [see Fig. 2(c), case(i)], there is40 an , 15% symmetric contribution to the magnetic anisotropy for an applied voltage of 2 V. Clearly the intrinsic Rashba field BR is modified when the materials adjacent to a 3d ferromagnet (F) are different. In a number of experiments an insulator I, often MgO, lies to one side and a normal metal (N), e.g., Au, Pt, Pd, Ta, or Ru, ‘‘cap’’ completes a tri-layer system. The potential, Figs. 2(d) and (e), will increase in passing from Fe to MgO but in passing from the F to N-layer the potential will either increase, Fig. 2(d), or decrease, Fig. 2(e). As discussed above, the relevant potential would be expected to increase, Fig. 2(d), in passing from a 3d element, e.g., Co, Fe, Ni, or Cu, to a 4d transition metal such as Ru, Pd, Ag but decrease, Fig. 2(e), for the 5d elements, e.g., Au, Pt, Ta. The latter case is particularly favourable since the intrinsic Rashba fields have the same sense and add. In addition, the 5d elements have a larger spin-orbit coupling, resulting in a larger aR and hence are more likely to produce a sizeable PMA. If the electric field decreases at the FI interface, the average Rashba field increases in the first case [Fig. 2(f)] when the effects of the surfaces tend to cancel and, as illustrated in Fig. 2(g), decreases in the second case when the inverse is true [see Fig. 2c, case (ii) and (iii)]. Experiment12 indeed shows an opposite field dependence for such systems with Pd(4d) and Pt(5d) N-layers. That the sign of the electric field contribution to the PMA reflects the N-layer whereas the field is applied to the opposite surface between F and I supports the current Rashba model. This is in stark contrast with the popular surface doping model3,12, for which the effects of surface doping are limited by the
Discussion The resulting anisotropy energy can be very large. The work already cited24–27 on conducting but non-magnetic materials helps set the scales. The value of the scaling prefactor ER in Eq. (5) for the surface state of Au is , 3.5 meV or about 35 T in magnetic field units and very much larger than the typical , 1 T demagnetising field. If a Au film is polarised by contact with an ultra-thin ferromagnet, the second factor, 2T/J0S, in Eq. (5) for the field inside a Au surface layer can be quite large , 5 leading to a PMA and indeed ultra-thin Fe on Au does have such a PMA36,37. Ultra-thin ferromagnetic films in contact with, e.g., Ru, Pd, Pt, and Ta, etc., also are found to have a PMA6,12,38,39. Schematically shown in Fig. 2(a) is the potential seen by electrons in a free standing ultrathin ferromagnetic film. At the surface, the potential reflects an electron’s image charge but reaches the vacuum level within a few atomic spacing. As already discussed, this results in a finite large electric field E , 10 V/nm at each surface but in opposite senses. Assuming an appreciable spin-orbit coupling in the interface region, this results, in turn, in a Rashba field BR which also changes sign between the two surfaces for a given momentum. Thus, for a perfectly symmetric film, the ferromagnetically polarised electrons see no average field BR. This symmetry can be broken by the application of an external electric field as shown in Fig. 2(b). The
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Figure 2 | (a) There is an electric field E in the surface region of a ferromagnet, however for a given wave vector k, the Rashba field BR, proportional to k 3 E, has an opposite sign at the two surfaces and the average field is zero. (b) With a finite external field this symmetry is broken and there is a net Rashba field acting upon the electrons. (c) The gate voltage dependence of the anisotropy energy. The internal electric field causes the shift of the parabola in the lateral axis as indicated by V0 for case i). For cases ii) and iii) the internal field shift is far beyond the external field range and nearly linear E-dependence arises. (d) The symmetry is also broken for a insulator-ferromagnet-metal sandwich. Also despite the electric field being smaller at the right surface, for a suitable metal, the spin-orbit coupling is larger and hence the metal interface can still dominate the net Rashba field. (e) Here the work function is larger for the metal than for the ferromagnet and the field for that surface is reversed. Now the Rashba fields at the two surfaces add. (f), (g) Applying positive gate voltages decreases the Rashba field at the insulating surface which, for this case, causes a net increase/decrease in the average Rashba field. SCIENTIFIC REPORTS | 4 : 4105 | DOI: 10.1038/srep04105
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www.nature.com/scientificreports (possibly magnetically modified) Fermi-Thomas screening length. In reality12, the screening length is estimated to be much less than 1 nm, and much too short for there to be an appreciable doping effect of the Pt or Pd layers that are typically distant by a few nanometers. Simulating a large applied electric field E, the required asymmetry might be controlled in NFN tri-layers by varying in a systematic manner, at the mono-layer level, the thickness of one of the normal metal layers and by using metals with different spin-orbit couplings. In reality the effect of the substrate transmitted to, and through, the bottom normal metal will imply an asymmetry even for largish Nlayer thickness. Indeed the PMA surface term for Au/Fe(110)/ Au(111) structures does show an non-monotonic dependence on the top Au layer thickness36. Experiments37 for Fe layers on vicinal Ag(001) and Au(001) surfaces and which undergo a symmetry breaking (5 3 20) surface reconstruction manifest an in-plane surface term reflecting this broken symmetry and which is larger for Au, with its stronger spin-orbit coupling, than for Ag. It is predicted that the surface coercivity field Hc is proportional to (Eint 1 Eext)2 where Eint is the internal electric field corresponding to the zero-bias Rashba contribution to the anisotropy. Such a nonlinear field dependence is observed, e.g., for the in-plane contribution for a (Ge,Mn)As/ZrO2 surface3. In other experiments10 with CoFeB/ MgO/CoFeB structures there is qualitative difference between the E dependence of the anisotropy field Hc of the, ‘‘top’’ and ‘‘bottom’’, CoFeB layers of this three layer structure, even when they have similar thicknesses. The bottom layer has a larger Hc and is roughly linear while Hc becomes highly non-linear as Hc R 0 as would be expected as Eext R 2Eint. The most direct experimental test of the model is the observation of the band splittings for a model Rashba system with a variable contact with an itinerant ferromagnet. This can result in giant magnetic anisotropy (GMA) energies. For example an ER , 100 meV (or , 1000 T) is reported in angle-resolved photoemission spectroscopy (ARPES) measurements41 on bulk BiTeI. For a thin film of this, or similar material, in contact with an itinerant ferromagnet such as Fe, a suitable exchange splitting J0S, tuned to the order of ER, might be induced and a GMA will result. ARPES performed as a function of the direction of the magnetisation m might determine both ER and the momentum dependence of the exchange splitting leading to estimates of both the PM and DM contributions and which might be directly compared with magnetisation and magnetic resonance measurements. The electrical control of such a GMA has evident important application for non-volatile memory applications. There are clearly many more complicated embodiments of such a device. In conclusion, it is suggested that the Rashba magnetic field due to the internal electric field in the surface region of an ultra-thin ferromagnet can make an important contribution to the perpendicular magnetic anisotropy. Such surface fields might be modified by application of an applied electric field. Since the internal fields at two surfaces tend to cancel, an asymmetry between the surfaces is important. Such an asymmetry is caused by different metal and insulator caping layers. These ideas are consistent with a large number of experiments.
Methods The non-magnetic case. This corresponds to Eq. (1) with J0 5 0. It is solved by taking the axis of quantisation ^z|k tobeperpendicular to the in-plane k as in Fig. 1(b). The eigenstates are eik?rjsæ and H~ h2 2m k2 {gmB BR Ns=2, where the Rashba magnetic field in energy units is defined as gmB BR ~2aR {ky ^xzkx ^y , with mB the Bohr magneton and g the g-factor, leaving the spin state jsæ to be determined. There are two concentric The energy splitting 2aR k:Dðk=kF Þ, where D is the value Fermi surfaces. for kF : kF: zkF; 2, with kF",# the Fermi wave number for the spin up/down (s 5 61) band. For the surface state of Au, D < 110 meV while EF < 420 meV giving the ER < 3.5 meV cited in the text. The magnetic case. The full Eq. (1) is solved by ^ defining axes such that m:S=S lies in the y–z-plane and S~Sðcos h^zzsin h^yÞ. The ^ total field, which defines the axis of quantisation, gmB BT ~2½ðJ0 SzaR kx sin hÞm{ ^ xÞ. It is assumed that, for a 3d ferromagnet J0S , 0.5–1.0 eV xzaR kx cos hðm|^ aR k y ^ and gmBBR , J0S, i.e., the Rashba is smaller than the exchange splitting. To second
SCIENTIFIC REPORTS | 4 : 4105 | DOI: 10.1038/srep04105
^ where JS~ ðJ0 SÞ2 za2R kx 2 cos2 hz order in gmBBR, gmB BT
